For \(z \in V_{p_0}\) . The gradient \(\text{grad}_pd(p,z)\)  for \(p \in U_{p_0}\)  is the velocity of the distance minimizing unit speed geodesic from \(z\)  to \(p\)  (see for instance [1]}). In particular the map
\(\tilde{H}_{p_0} \colon (V_{p_0}\cap \Gamma ) \ni z \rightarrow \text{grad}_pd(p_0,z)\in S_{p_0}M\) 
